diff --git a/approx_alg.tex b/approx_alg.tex
index 469280e..7eba36e 100644
--- a/approx_alg.tex
+++ b/approx_alg.tex
@@ -2,7 +2,7 @@
 %!TEX root=./main.tex
 
 \section{$1 \pm \epsilon$ Approximation Algorithm}\label{sec:algo}
-In \Cref{sec:hard}, we showed that \Cref{prob:bag-pdb-poly-expected} cannot be solved in $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$ runtime.  In light of this, we desire to produce an approximation algorithm that runs in time $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$.  We do this by showing the result via circuits,
+We showed in~\Cref{sec:hard} that a runtime of $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$ cannot be acheived for~\Cref{prob:bag-pdb-poly-expected}.  In light of this, we desire to produce an approximation algorithm that runs in time $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$.  We do this by showing the result via circuits,
 such that our approximation algorithm for this problem runs in $\bigO{\abs{\circuit}}$ for a very broad class of circuits, (thus affirming~\Cref{prob:intro-stmt});  see the discussion after \Cref{lem:val-ub} for more.
 The following approximation algorithm applies to bag query semantics over both
 \abbrCTIDB lineage polynomials and general \abbrBIDB lineage polynomials in practice, where for the latter we note that a $1$-\abbrTIDB is equivalently a \abbrBIDB (blocks are size $1$).  Our experimental results (see~\Cref{app:subsec:experiment}) which use queries from the PDBench benchmark~\cite{pdbench} show a low $\gamma$ (see~\Cref{def:param-gamma}) supporting the notion that our bounds hold for general \abbrBIDB in practice.
@@ -21,7 +21,7 @@ $\expansion{\circuit}$ has the following recursive definition ($\circ$ is list c
 $\expansion{\circuit} =
 \begin{cases}
 					\expansion{\circuit_\linput} \circ \expansion{\circuit_\rinput}		&\textbf{ if }\circuit.\type = \circplus\\
-					\left\{(\monom_\linput \cup \monom_\rinput, \coef_\linput \cdot \coef_\rinput) ~|~(\monom_\linput, \coef_\linput) \in \expansion{\circuit_\linput}, (\monom_\rinput, \coef_\rinput) \in \expansion{\circuit_\rinput}\right\} 		&\textbf{ if }\circuit.\type = \circmult\\
+					\left\{(\monom_\linput \cup \monom_\rinput, \coef_\linput \cdot \coef_\rinput) \right.\\\left.~|~(\monom_\linput, \coef_\linput) \in \expansion{\circuit_\linput}, (\monom_\rinput, \coef_\rinput) \in \expansion{\circuit_\rinput}\right\} 		&\textbf{ if }\circuit.\type = \circmult\\
 					\elist{(\emptyset, \circuit.\val)}								&\textbf{ if }\circuit.\type = \tnum\\
 					\elist{(\{\circuit.\val\}, 1)}									&\textbf{ if }\circuit.\type = \var.\\
 \end{cases}
@@ -102,11 +102,13 @@ satisfying
 \probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error' \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf
 \end{equation}
  can be computed in time
+\begin{footnotesize}
 \begin{equation}
 \label{eq:approx-algo-runtime}
 O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right).
 \end{equation}
-In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$.
+\end{footnotesize}
+In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $$O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right).$$
 \end{Theorem}
 
 The restriction on $\gamma$ is satisfied by any
diff --git a/main.pdf b/main.pdf
index 75807dc..5542b23 100644
Binary files a/main.pdf and b/main.pdf differ
diff --git a/main.synctex.gz b/main.synctex.gz
index f8f207c..2ec2e3d 100644
Binary files a/main.synctex.gz and b/main.synctex.gz differ